A lot of people are FANATICAL about tank size, and some fish, like goldfish really do need large tanks because they produce ALOT of waste. However, bettas aren't that messy and in MY opinion as long as the water is heated and weekly water changes are made, 1 gallon is OK. Personally, I like to give my bettas at least 3 gallons each - more room for them. If you go to PETA's webpage they insist that one betta MUST have 10 gallons...
Cycling isn't as bad as it sounds. I put off having filters for a long time for my 10 and 20G tanks but in the long run, it makes things alot easier on you. Basically, all you have to do is keep the filter running and test the water every 3 or so days. Keep an eye on the ammonia, nitrite and nitrAte levels. If any of them get to high, do a 50% water change. First the ammonia will rise and drop off, then the nirItes will do the same and then the nitrAtes will come along. When you get nitrAtes but no ammonia or Nitrites, you are cycled. You want to keep the nitrAtes under 40, I think. Ammonia should be at 0 but you need to have some of it in your tank for the ammonia eating bacteria. I changed my water when it got to 1.0 as I have ammonia in my tap so it will always read .25. If you by the API master test kit - it will be easy to tell when to change the water
clamped fins do mean something is wrong but it could be anything - like being too cold, the water is too dirty or possibly there is too much open space and he feels vunerable. It does not necissarly mean the fish is sick but almost all sick fish will be clamped.
with 6 gallons, you can divide the tank and have 2 bettas or you can add a snail or some shrimp. However snails are poop machines and shrimp may become snacks for the betta. Plus some snails are asexual and reproduce like mad, while others need a male and female. I think snails are neat :)
Ditto on not rinsing the filter one you have it in the tank. The bacteria you want will colonize in the filter. They also live in the gravel and on any decor in the tank so you shouldn't do too much cleaning unless its an emergency. ALso, what filter do you have? if it has the removable cartridges that are supposed to be replaced ever 4 weeks, do not replace them. Just swish them around in old tank water and put them back in the filter. Most of the bacteria you want will be on them and if you replace the cartridge, there goes the majority of your helpful bacteria and your cycle. This may or may not be true depending on the filter you have. Some, like the marineland penguine have a biowheel that holds the bacteria and aqua clear ones have several different types of filter media that can be removed.